Problem: Suppose we have a vector-valued function $g(t)$ and a scalar function $f(x, y, z)$. Let $h(t) = f(g(t))$. We know: $\begin{aligned} &g(7) = (4, 5, 0) \\ \\ &g'(7) = (15, 6, 5) \\ \\ &\nabla f(4, 5, 0) = (2, -5, 1) \end{aligned}$ Evaluate $\dfrac{d h}{d t}$ at $t = 7$. $h'(7) =$
Formula The multivariable chain rule says that $\dfrac{dh}{dt} = \nabla f(g(t)) \cdot g'(t)$. The $g'(t)$ part is how much a change in $t$ will cause the input to $f$ to move, and the $\nabla f(g(t))$ part is how much $f$ will change in response to this update to its input. [What's the intuition behind the formula?] Applying the formula We want to find $h'(7) = \nabla f(g(7)) \cdot g'(7)$. We know the following. $\begin{aligned} &g(7) = (4, 5, 0) \\ \\ &g'(7) = (15, 6, 5) \\ \\ &\nabla f(4, 5, 0) = (2, -5, 1) \end{aligned}$ Substituting: $h'(7) = (2, -5, 1) \cdot (15, 6, 5) = 5$ Answer Therefore, $h'(7) = 5$.